Question

Find the direction of the axis of symmetry of the quadratic surface $$ 7 x^{2}+7 y^{2}+7 z^{2}-20 y z-20 x z+20 x y=3 $$

Short answer

The direction of the axis of symmetry is along the vector \( (-1, -1, -1) \).

Step-by-step solution

Identify the Given Equation

The given quadratic surface equation is: \[ 7x^2 + 7y^2 + 7z^2 - 20yz - 20xz + 20xy = 3 \]

Normalize the Coefficients

Divide the entire equation by 7 to simplify the coefficients: \[ x^2 + y^2 + z^2 - \frac{20}{7}yz - \frac{20}{7}xz + \frac{20}{7}xy = \frac{3}{7} \]

Rewrite in Matrix Form

Express the quadratic form in matrix notation: \[ [x \, y \, z] \begin{pmatrix} 1 & \frac{10}{7} & -\frac{10}{7} \ \frac{10}{7} & 1 & -\frac{10}{7} \ -\frac{10}{7} & -\frac{10}{7} & 1 \end{pmatrix} \begin{pmatrix} x \ y \ z \end{pmatrix} = \frac{3}{7} \]

Find Eigenvalues and Eigenvectors

Solve the eigenvalue problem for the matrix \[ \begin{pmatrix} 1 & \frac{10}{7} & -\frac{10}{7} \ \frac{10}{7} & 1 & -\frac{10}{7} \ -\frac{10}{7} & -\frac{10}{7} & 1 \end{pmatrix} \]. The eigenvalues are found to be: \[ \lambda_1 = 3, \lambda_2 = -1, \lambda_3 = 3 \].

Identify Corresponding Eigenvectors

Find the eigenvectors corresponding to the eigenvalues: - For \(\lambda_1 = 3\), the eigenvector is \( \begin{pmatrix} -1 ewline -1 ewline -1 \end{pmatrix} \) - For \(\lambda_2 = -1\), the eigenvector is \( \begin{pmatrix} 1 ewline 1 ewline 1 \end{pmatrix} \) - For \(\lambda_3 = 3\), the eigenvector is \( \begin{pmatrix} 1 ewline 0 ewline -1 \end{pmatrix} \).

Determine the Axis of Symmetry

The axis of symmetry corresponds to the eigenvector associated with the repeated (and larger) eigenvalue. Thus, the axis of symmetry is the eigenvector corresponding to \(\lambda_1 = 3\): \( \begin{pmatrix} -1 ewline -1 ewline -1 \end{pmatrix} \).

Key concepts

eigenvalues

Eigenvalues are crucial in understanding the properties of quadratic surfaces. For a given matrix, eigenvalues are the special set of scalars associated with the matrix, such that when you multiply the matrix by a certain non-zero vector (the eigenvector), the product is the same as multiplying that eigenvector by the scalar (the eigenvalue).

In mathematical terms, for a square matrix A, an eigenvalue \( \lambda \) and its corresponding eigenvector \( \mathbf{v} \) satisfy:
\[ A \mathbf{v} = \lambda \mathbf{v} \]

This relationship helps in transforming the quadratic equation into a more manageable form. In our problem, solving for the eigenvalues helped us determine important characteristics about the quadratic surface's symmetry properties.

eigenvectors

An eigenvector is a non-zero vector that changes only by a scalar factor when a linear transformation is applied. Essentially, it points in a direction that remains unchanged under the given transformation. For our specific quadratic surface problem, the eigenvectors correspond to the directions in space that define the axes of symmetry.

Given a matrix A and an eigenvalue \( \lambda \), the eigenvector \( \mathbf{v} \) meets the following condition:
\[ A \mathbf{v} = \lambda \mathbf{v} \]

This tells us that when matrix A operates on the eigenvector \( \mathbf{v} \), the direction of \( \mathbf{v} \) does not change, though its length is scaled by \( \lambda \). In our problem, solving for eigenvectors helped us to find the axis of symmetry associated with the quadratic surface.

matrix form

Rewriting a quadratic form in matrix form simplifies solving and understanding the problem. A quadratic equation in three variables, like the one we are dealing with, can be represented using a symmetric matrix. This representation allows us to apply linear algebra techniques.

The given quadratic surface:
\[ 7x^2 + 7y^2 + 7z^2 - 20yz - 20xz + 20xy = 3 \]

can be rewritten in matrix form as:
\[ [x \ y \ z] \begin{pmatrix} 1 & \frac{10}{7} & -\frac{10}{7} \ \frac{10}{7} & 1 & -\frac{10}{7} \ -\frac{10}{7} & -\frac{10}{7} & 1 \end{pmatrix} \begin{pmatrix} x \ y \ z \end{pmatrix} = \frac{3}{7} \]

This matrix representation encapsulates the relationships between the variables in a compact form, and it is essential for performing eigenvalue and eigenvector analysis.

axis of symmetry

The axis of symmetry of a quadratic surface represents a line that the surface 'reflects' upon, remaining unchanged in its reflection. For a quadratic surface like the one given, determining this axis involves finding the eigenvectors that correspond to the eigenvalues of the associated matrix.

In our exercise, we found the eigenvalues \( \lambda_1 = 3, \lambda_2 = -1, \lambda_3 = 3 \) and their corresponding eigenvectors. Since the axis of symmetry corresponds to the eigenvector associated with the repeated and larger eigenvalue, the eigenvector matching \lambda_1 = 3 \ was identified:
\[ \begin{pmatrix} -1 \ -1 \ -1 \end{pmatrix} \]

This vector points in the direction of the axis around which the surface symmetrically aligns. Understanding the axis of symmetry helps visualize and comprehend the geometric shape and behavior of the quadratic surface.